DDSS2006　

A Comparison Study about the Allocation Problem of Undesirable Facilities Based on Residential Awareness

– A Case Study on Waste Disposal Facility in ChengDu City, Sichuan China –

K. Zhou, A. Kondo, A.Cartagena Gordillo1 and K. WatanabeThe University of Tokushima, Tokushima, Japan 1Yokohama National University, Yokohama, Japan

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Motivation I

• In the existing facility location theory, there are many studies concerned with location modeling of facilities that put more importance on nearness. But there are also some facilities that give undesirable feeling to residents. Location problems for those kind of facilities require new methodologies with corresponding solutions.

• Thereupon, in this research, our purpose is to consider the location problem of undesirable facilities

• Waste disposal facilities are appointed for analysis. Specifically, we choose garbage transfer stations and final disposal facilities as research objects due to their high level of variety.

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Motivation II

• As we review the methodologies for location problems of undesirable facility, we found that the most popular way of handling undesirability for a single facility is to minimize the highest effect on a series of fixed points applying the principle of locating the undesirable facilities as far as possible from all sensitive places.

• Therefore, in the existing literature we can appreciate that physical magnitudes, such as distance or time, were mainly used as important parameters on the study of facility location. However, the psychology element of the facility users was not given enough attention.

• Regarding those characteristic, our objective in this research is to analyze location problems of undesirable facilities by using a model based on probability theory, which considers residential awareness.

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Contents

3-Parameter LoglogisticWeibull Non-Parametric

Regression Analysis Matlab Minitab

Parameters Estimation

Endurance Rate Function

Conclusions

4

Stochastic Methods

Modeling

5

Definition of Endurance Distance & Endurance Rate

For purely undesirable facilities, we can consider that residents hope the undesirable facility can be located farther than a certain distance, which means the residents can endure the location of the undesirable facility if the facility is located farther than that distance. Then the minimum of this desired distance can be defined as endurance distance, which is expressed here as w. And, when an undesirable facility is located at a certain distance, the rate of residents who could endure the facility location is defined as endurance rate, which is expressed here as P(x) in this research.

●

Residential location

Undesirable facility

Endurance distance

W

X > W (X : distance to a facility)

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6

Distribution of Endurance Distance

)1(0.1)(0 dwwf

)2()()(0x

dwwfxP

Fig.1 Relationship between the endurance rate and distance to a facility

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Assumption for Distribution of Endurance Distance

)4(1)(0

1

mm xx

wm edwew

mxP

)3()( 1

mwm ew

mwf

)5())(1(log

m

e

xxP

)6(1

loglog))(1(loglogeeee xmxP

where α and m are scale and shape parameters. Y mX c

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Survey Concerning Endurance Distance

Estimation of parameters for endurance rate function

Data concerning the endurance rate P(x)

Carry out a questionnaire survey toward the residents in object area

Questionnaire Survey

At least how far should a waste facility be located to your home?

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9

Questionnaire Survey in Chengdu City

Fig.2 The location of Chengdu City

Chengdu

Beijing

Shanghai Chengdu

Beijing

Shanghai

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10

Case Study Area-object of Survey

― ― ：Main Road

― ― ：other Road

Survey Area

Fig.3 Object area of the research

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11

The Result of Survey

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3 % 4 %

11 %

3 1 %

2 4 %

2 7 %

1 0 's

2 0 's

3 0 's

4 0 's

5 0 's

6 0 's ～

Fig.4

Percentage by age

Distributed survey

Valid answers

Valid response rate

350

323

Percentage by sex

92.3%

male

52%

female

48%

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18.0

0.3 0.6

44.3

23.5

13.3

0

20

40

60

0～3 4～6 7～9 10～12 13～15 16～Distance (km)

Per

cent

age

of P

eopl

e (%

)

Fig.5

Distance to garbage transfer

stations and corresponding

residential endurance rate

5.6 8.0 6.5

56.7

6.5

12.7

4.00

20

40

60

80

1～5 6～10 11～15 16～20 21～25 20～30 31～

Distance (km)

Per

cent

age

of P

eopl

e (%

)

Fig.6

Distance to final waste disposal facilities and corresponding residential endurance rate

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Endurance Distance and Endurance Rate

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Average Endurance Distance Classified by Attribute

5.867.28 7.56 7.83 7.17

9.73

0

3

6

9

12

teens twenties thirties forties fifties sixties andabove

km

Fig.7 Average endurance distance classified by age for garbage transfer stations

25.9329.66

35.08

25.66

32.2727.75

0

10

20

30

40

teens twenties thirties forties fifties sixties andabove

km

Fig.8 Average endurance distance classified by age for final waste disposal facilities

7.42

31.13

7.64

28.58

0

7

14

21

28

35

garbage transfer stations final waste disposal facility

km

male

female

Fig.9 Average endurance distance classified by sex

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Estimated Parameters of Endurance Rate Function

garbage transfer final waste disposal

stations facilities

parameter α 38.7 (9.971) 522.5 (19.312)

parameter m 1.809 (9.383) 1.781 (16.968)

R 2 0.880 0.947numbers of sample 14 18

facilities

Table 1. Result of parameters estimation for the endurance rate model

where the numbers between parentheses represent the value of t.R2 is determination Coefficient

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15

0

20

40

60

80

100

0 5 10 15 20

Distance (km)

End

uran

ce R

ate

(%)

curve from the model plots from survey data

Fig.10 The residential

endurance rate for garbage transfer stations

0

20

40

60

80

100

0 20 40 60 80Distance (km)

End

uran

ce R

ate

(%)

curve from the model plots from the survey data

Fig.11 The residential

endurance rate for final waste disposal facilities

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NON-PARAMETRIC DISTRIBUTION METHOD

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In matrix form, non-linear models are given by the formula:

y = f(X, β)+ ε,

where y is an n-by-1 vector of responses, f is a function of β and X, β is a m-by-1 vector of coefficients, X is the n-by-m design matrix for the model, ε is an n-by-1 vector of errors, n is the number of data and m is the number of coefficients.

The fitting process was automated, employing the commercial software Fitting Toolbox from Matlab.

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Distribution Fitting for Non-parametric Distribution

a1 = 0.0001 a2 = -1.41 a3 = 0.323 a4 = 0.040 a5 = -0.237

b1 = -1.62 b2 = 5.108 b3 =10.00 b4 = 7.935 b5 = 6.962

c1 = 56.90 c2 = 2.364 c3 = 0.682 c4 = 0.565 c5 = 1.628

a6 = 1.541 a7 = -0.222 a8 = 0.067

b6 = 5.359 b7 = -45.73 b8 = 1.015

c6 = 2.599 c7 = 24.950 c8 = 0.744

SSE: 0.0002

R2: 0.9997

RMSE: 0.0012

Table 2. The coefficients of equation (7) and goodness of fit

where, y(x) is probability distribution function (7)2^8/)8((exp(82^7/)7((exp(7

2^6/)6((exp(62^5/)5((exp(52^4/)4((exp(4

2^3/)3((exp(32^2/)2((exp(22^1/)1((exp(1)(

cbxacbxa

cbxacbxacbxa

cbxacbxacbxaxy

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Cumulative Distribution Function Calculation

d1 = 0.64e-1 d2 = 0.44e-1 d3 = -2.96 d4 = 0.195 d5 = 0.2e-1 d6 =-0.342 d7 = 3.549 d8 = -4.91

e1 = 0.18e-1 e2 = 1.344 e3 = 0.423 e4 = 1.466 e5 = 1.770 e6 = 0.614 e7 = 0.385 e8 = 0.401

g1 = 0.28e-1 g2 = -1.364 g3 = -2.161 g4 = -14.66 g5 = -14.04 g6 = -4.28 g7 = -2.06 g8 = 1.833

h1= 5.363

Table 3. The coefficients of equation (8) (with 95% confidence bounds)

1)88(8)77(7)66(6)55(5

)44(4)33(3)22(2)11(1

)()(0

hgeerfdgeerfdgeerfdgeerfd

geerfdgeerfdgeerfdgeerfd

dxxyh

where erf(.) is the error function (8)

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0

20

40

60

80

100

0 2 4 6 8 10 12Distance (km)

End

uran

ce r

ate

(%)

curve from the model plots from survey data

Fig.12 Distribution of the endurance rate h(λ)

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3-PARAMETER LOGLOGISTIC DISTRIBUTION METHOD

Analysis of Data

_Number of People

Distance_IntervalDensity Coefficient

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0

50

100

150

0 2 4 6 8 10 12

Distance (km)

Num

ber

of

Peo

ple

Fig.13 A plot of the original data to 12km

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Flipping the Data

Fig.14. Data flipping

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Flipping the Data

)12()( xfxg (9)

)12()( xhxj (10)

Fig.15 NewData

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Where f(x) is the distribution in Figure 9 , g(x) is the distribution in Figure 10.

0

50

100

150

0 2 4 6 8 10 12

Distance (km)

Num

ber

of

Peo

ple

Where h(x) is the distribution function for New Data,j(x) is the distribution function for original data.

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Distribution Analysis of NewData

2)ln(

)ln(

1)(

1)(

b

acx

b

acx

e

e

cxbxh (11)

where, a = Location parameter, b = Scale parameter, c = Threshold parameter

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Employing the estimation method of Least Squares, a 3-Parameter Loglogistic distribution function was found as:

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The Resulting Values for the Parameters & the Goodness of Fit

C1　- 　Threshold

Percent

100101

99.9

99

959080706050403020105

1

0.1

Table of Statistics

StDev 5.16988Median 3.56087IQR 2.95870Failure 320Censor 0

Loc

AD* 0.465Correlation 0.997

1.18332Scale 0.399455Thres 0.295683Mean 4.60651

Probability　Plot　for　C1

Complete Data - LSXY Estimates3- Parameter Loglogistic - 95% CI

Fig.16 Result of parameter estimation and test of goodness of distribution　 (Where, C1 means NewData shown in Figure 13)

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a=1.183

b=0.399

c=0.296

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2)12(ln

)12(ln

1)12(

1)(

b

acx

b

acx

e

e

cxbxj

265.19)7.11(

)7.11(04.01)()(

2

5

2

5

0

z

zdxxjzsr

z

(12)

(13)

where z is a value between 0 and 12, 0.04 is the integral of the resulting function from -∞ to 0.

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The modified Loglogistic function is:

Finally, the function of the endurance rate model is:

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04.196.0

1

)(

1

)()(

max

zsrCC

zsrCCznewsr

(14)

0

20

40

60

80

100

0 2 4 6 8 10 12Distance (km)

En

du

ran

ce r

ate

(%)

curve from the model plots from survey data

Fig.17 Distribution of the endurance rate sr(z)

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CONCLUSIONS

• Regarding undesirable facilities, we defined residential endurance distance and endurance rate, modeled the relationship between facility’s location and the endurance rate.

• From the questionnaire survey carried out in Chengdu City, we could dissect the distribution of residential endurance distance for garbage transfer stations and final waste disposal facility. Using the endurance rate model, we indicated it’s possible to propose waste facilities’ location from the viewpoint of residents.

• Based on different probability distribution functions, we proposed three models for estimating the residential endurance rate and make a comparison study. Based on those models, we found there’s no big difference between the results when residential endurance rate according to facility location is 80%, the calculation results of suitable distance for garbage transfer stations are all around 10km.

• From the comparison study, we found that the advantage of the model employing Weibull distribution is its simplicity; it has only 2 parameters and can be used for the both 　 kinds of facilities though the accuracy was not good enough. The Non-parametric one described a better modelling even though a lot of parameters were needed for describing the detail of the data. As computer technology is developed today, we consider that this method can be used for any kind of situation as a numerical analysis model. Based on a parametric distribution function, we also found a model by analysing and flipping the data as explained above. For this case, the model using Loglogistic distribution function is a new experiment with good modelling characteristics.

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Thank you for your attention!

質　問　対　策

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Slide 3

• What’s the meaning of “the highest effect”?

• What are the meaning of “fixed points”?

• What means “a series of fixed points”?

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Slide 3

• Why need consider residential awareness in this study?

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Why & how could find Weibull

During the proceeding, we found Weibull distribution has some interesting characters as following: 1. It is a distribution with good elasticity. The shape changes following shape parameter’s changing. 2. The distribution function is completely integrabel, which make it possiblefor next step of parameter estimation.

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Contents of the survey

For getting the data, a survey on residential awareness about undesirable facilities was carried out in Feb. 2004. The area object of survey is shown in Figure 3. The question was: At least how far should a waste facility be located to your home? According to the endurance distance, a few alternatives were given in advance. Then respondents choose their desired endurance distance from the alternatives or a certain number they considered adequate. The choices, for garbage transfer stations, were from 1km to 10km, for final waste disposal facilities, were from 5km to 30km. For both facilities there was the option: “If there’s no endurance distance you considered, please write down a distance you can endure”. Data analysis was based on the endurance distance which residents chose or wrote. A simple explanation concerning present condition of waste disposal in Chengdu was given before the questions.

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What means

In matrix form, non-linear models are given by the formula:

y = f(X, β)+ ε,

where y is an n-by-1 vector of responses, f is a function of β and X, β is a m-by-1 vector of coefficients, X is the n-by-m design matrix for the model, ε is an n-by-1 vector of errors, n is the number of data and m is the number of coefficients.

The fitting process was automated, employing the commercial software Fitting Toolbox from Matlab.

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Table 2

If the value of “goodness of fit” can be gain at the step of distribution fitting?

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The procedure of selecting equation (11) by employing MINITAB

37

Explaining the following paragraph

The function j(x) in equation (13) exists for values x<0, which is unreal for the processed data, then an adjusting value of 0.04 is included in equation (13) which corresponds to the integral of j(x) from -∞ to 0. For this reason, the endurance rate never reaches 100%. A corrective coefficient can be applied to the equation of the endurance rate sr(z). Then it becomes equation (14). The corrected function newsr(z) is illustrated in Figure 12.